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 Définition d'une variable
  
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Appendix D
What to do in School and Why

Previous:  C How to Read

A  Why Go or Attend

Education is for both sons and daughters. The assumption that someone else will provide home and shelter for us is too often false. So you should get the most out of your schooling and education. Your ability to get a job, your future earnings and your future joys and companions may all depend on what you learn in and after school. All this may also depend on factors beyond your control (wars, social conditions, the economy or parents). The suggestions provided below are for some, not all eventualities.

School can be a place where you learn to master rule and pattern-based thought (logic) and where you master further skills and knowledge. You are in school to learn about the wide range of human knowledge and behavior. Try to understand whatever you might be asked to study or do. Think for yourself and ask why you or others do this or that.

In school, you should look for the ideas new to you and for the ideas worth repeating to others. That is what makes a class or a subject worthwhile. Again, ideas which you have already seen are reassuring and comforting - not much work is needed here, but you only learn from the ideas new to you. Look and search for them.

What you see in school and in books represents the experience of others. When you are observant, you can learn from the experience, skills and mistakes of others instead of your own.

History courses, English (or literature) courses, science courses and some realistic novels offer this experience secondhand. Secondhand knowledge of hardship is preferable to first hand. Ask your teachers, relatives and the others about the joys and the difficulties they met or foresee. Learn from their experience. Ask for opinions. Guidance from others requires the statement of opinions. Get two or more opinions even if you liked the first. Seek and politely allow opinions different from your own. Different points of view may sharpen or change yours.

B  Health and Social Skills

You are also in school to meet people and to learn how to mix or socialize with others. To this end, join a club or group activity. See how people, including yourself behave in groups. After school, the opportunities to mix may decline - or you may not develop the habit of mixing and socializing. Choose activities you like. Try one, two or several, but leave enough time for your studies and for special events. Suggestion: each week, get three or more hours of physical exercise. This exercise could come from physical labor. Or, you may find a sport or activity which you can do now and later. This exercise should build your health without risking it and without damaging it.4

4I saw in a university soccer match or practice in 1984, a player with a small cast on his leg. I thought he was risking permanent damage.

In 1990, after twenty years of cross-country skiing for exercise etc, I began to ski in colder and colder conditions without fear. The eventual result was a deep frostbite to my cheeks, an area difficult to protect. Then for five years, exposure to the cold was an unpleasant experience followed by hours of pain or discomfort - a distraction from work and play that is not recommended.

C  Suggestions for Learning

By law you are required to attend school. Make sure your time is not wasted. Make sure that some of your courses are with helpful, hard-driving, teachers. Ask them for advice on what to do or what could be useful to you. Further advice follows. It repeats in part advice given in previous appendices.

  1. Look for the ideas new to you and for ideas worth repeating. When you are preparing for a test or for a future lesson, your studying is done when you can find no ideas new to you.
  2. Try to remember the names of places, people and ideas. You can use the names in conversations, essays and tests later.
  3. Learn to read precisely what is written. This skill will serve you well. It gives you more independence both in class and when you leave school. It may allow you to learn at your own pace.
  4. Learn to take notes. When no textbook is present, note-taking skills will be needed. When a textbook is present, look at it first. (Reading it in advance may allow you to take fewer notes and understand lessons better.)
  5. Learn to type. Today, computers are used in all areas of office work bureaucracy and technology. These computers are controlled by keyboards. Accurate, if not fast, typing skills will make your exposure to computers and report writing more pleasant.5

    5This advice is valid now. Advances in computer technology - the introduction of voice-controlled dictation/computer systems - may make part of this advice stale or obsolete.
  6. Get careful thinking skills. That is, master the use of rules and patterns. Every area of skill and knowledge offers rules and patterns which you might follow. Learn to read exactly what each says. To follow, to agree or to disagree with rules, you need to understand exactly what they say and exactly what they don't say.
  7. In high school take courses which provide immediate job skills such as auto-mechanics, typing, metalwork, woodwork, drafting etc. Master arithmetic and learn to read and write carefully. Employers want skilled workers. They are easier to train and worth keeping. Even if you are planning a college education, practical job skills could get you a summer job. They may allow you to work and pay for part of a college education. Care is required to take the best and avoid the worst of the academic and non-academic courses in your school.
  8. Take English or master another language of your choice, well. This includes reading, writing, speaking and reasoning. When you write, tell a story, describe what is, or present an opinion or defend one. Watching for ideas worth repeating, will help.
  9. Take history courses. Courses with ideas new to you are worth taking. If possible, avoid history courses which only promote the group, state or country in which you live. History courses tell us about the experiences and mistakes of past, if not present generations.
  10. Read newspapers which do not (always) glorify the nation or group to which you belong. Contrary opinions make us think. So look for and read newspapers with views you occasionally find disagreeable.
  11. A little uncertainty in the words of a teacher leaves room for thought and the practice of thinking skills.

 


Appendices with (repetitive) advice for Students: B  How to Learn ] C. How to Read ] [ D. What to do in School ] PS. Study Tips ] PS: Time and Effort ] E. How to Study Math and Why ]

 

Three Skills
For 
Algebra
Volume 2
Printed in Canada
ISBN 0-9697564-2-9

Read slowly,  this work may enrich your skills & knowledge. Take the risk.

Chapters and Appendices

B  How to Learn
C. How to Read
D. What to do in School
PS. Study Tips
PS: Time and Effort
E. How to Study Math and Why

Foreword
1. Introduction
2. Implication Rules [4]
3. Chains of Reason [3]
4. Induction Mathematical
4. Romeo and Juliet
6  Old Language
5 Knowledge Islands [2]
7  Arith Skill Check [4 X 2]
Arith Webvideos
7. The Next Chapters
8 The Three Skills
8 VNR-Concise-Encyclopedia
PS. What is a Variable [8]
9. Algebra Talk [7]
10 Two More Skills[5]
11 Why Shorthand
12 Shorthand Usage [10]
13 What's Next
PS: The 4-th Skill For Algebra
14 Compound Interest [6]
15 Linear Equations [5]
16 Painless Proofs
17 Pythagoras
PS I.  Distributive Law
PS II. Polynomials
18 Rules of Algebra [20]
19  Functions & Sets
20 Degrees & Radians
21 What's Next
22. Arith & Geometric Sums [2]
23 Summation Notation
24 Your Money [3]
25 Induction & Recursion [4]
26 What's Next
27 Pronouns in Logic
28 Occurrence Tables
29 Contrapositive
30 Truth Tables
31 Indirect Reason
Pathways for Learning

 

What is a Variable?
Introduction
Variation between Examples

Variation of Letters

A letter denotes a variable

Cases of Double Variation

Three Notions of a Variable

Constants, Parameters
& Variables

Talking about numbers
Dependent or Independent
Variable, a Matter of Choice

For Senior High School  & Calculus Students

  <| (o)   (o)   |> 
 \     | |      / 
\___ _/

||
 -/[]\- 
||
   / \_ 

Words  to clearly introduce algebra and variables have been missing in course design. For people who cannot do algebra, 
the missing words may explain or ease their difficulties.  Volume 2 ,Three Skills for Algebra,  in Chapters 8 to 14 & 18 etc, puts words before symbols to providing the missing words in a way that enrich the comprehension of all.  Those words form the middle part of a algebra (and logic) lessons aimed at helping or improving all of  high school mathematics and also calculus course design & delivery. 

For Avid Readers in School & Out - Online Books 
   1.  Elements of Reason. 1996 
1A. Pattern Based Reason  1995 
1B. Math Curriculum Notes 1996 
2. Three Skills for Algebra  1995 
3.
Why Slopes & More.Math 1995
Tour their 
forewords.   

Calculus Prep or Help: See Volumes 2 & 3, and this bigger Calculus Guide.  If your  calculus   questions is not answered here, submit it. Over time, that may complete the site development of calculus. 

For Parents: Speaking Skills, Reading & Writing Preparing for Scienceends, values and methods for work and study,  parent- friendly maths skill development booklets for ages 4-14.

Mostly For High School

Intro to Solving Linear Equations
 
- a different paths for junior and even senior high school students. Question for Tutors: When do you use and when you skip the stick diagram method here?

Fraction Skills,  thought-based  development, Ages 10 to 14 may need a tutor.  Students who have to understand in order to do may like the development in all or part. 

For Senior High School Mathematics & Calculus

5
wordy Logic Chapters
4 curious Algebra Chapters
Words before & besides symbols. A Key Algebra forward & backwards Chapter   
 

First Calculus Preview (1st intro)
Four Calculus Chapters  (2nd intro)
Intro to Complex Numbers (long)
Intro to Mathematical Induction (romantic & wordy at first)

Tutors & Instructors: These lessons introduce skills differently Would you recommend them? 

More Topics 

1. Decimal Arithmetic  Reference!
2. Integers - Intro to Signed No.s

3.  Fractions - fully explained.
4.  Fractions  with Units  
5.   Number Theory
6.    Solving Linear Equations  
Formulas for- & backwards -  
8.  Proportionality, Back- & For-wards.   
9. Logic Chapters:   
10.  Euclidean-Geometry  
11.  Slopes & Equations of Straight Lines.  (Take I. See take II below)
12.  Why Study Slopes
13. Maps, Plans,  Similarity & Trig,  
  (Take II included here)
14.  Quadratics: Starter lessons
15.  Polynomials: Starter lessons 
16 Why Factor Polynomials:  
17   Functions - Forwards & Backwards.  
18.  Exponents, Radicals & logs.  
19
Complex Numbers before trig (new advance/ starter lesson)
20.  DC Electric Circuits Etc 
21.
Real  Analysis 
22. The Olde Complex No, Trig
& Vector Section.
23. More Calculus Stuff
- written after Volumes 2 and 3.

Level I Material: New Stuff
Time and Date Matters
Level I Arithmetic. 
Money Matters
Measurement Matters
Matters of Chance (Risk Control)
Logic Chapters (leave what's not clear in Level I to Level II)
Using/Making Maps and Plans.
(A variant of
Maps, Plans,  Similarity & Trig,  to appear here).

For Instructors
-
Education Essays   (opinions, possibilities, references) 
- Free Advice and Directions for teaching primary & high school maths will be given in online meeting place with voice & whiteboard.   
- Math & Logic  How-TOs 
1. Arithmetic
2. Algebra
3. More Algebra
4.  Beginner Geometry
5.  More Geometry
6. Calculus 
7. Show Work or Logic 
These may be too dense for students.

Offering ideas to change education makes this site different.  Nothing ventured, nothing gained.  Site material is mathematically  correct, and where not, please report errors. The two level program POMME in the site entrance implies multiple paths for instruction. Supporting those paths in turn implies a clear destination  for site development and perhaps a new name.


 


www.whyslopes.com > Volume 2,  Three Skills for Algebra  1995  >   D. What to do in School     Back ] Next ]


Road Safety Message   Walk on a side walk. If that is not possible, try  not to  walk on a road with your back to the traffic.
Try to see what  trucks, cars, buses or bicycles are coming, so that you may step out of their way.  Put safety first. .

Support for Technical Mathematics from Number Theory to Calculus Prep

A. More Arithmetic a must for algebra etc D. Logic In Mathematics G. Algebra with Take Home Value I. Vectors & Functions
Decimal Lesson - Reference  
Counting & Addition
   (8 lessons)
Comparison to Subtraction
  (9 lessons)
Multiplication
( 11 lessons)
Long Division  (12 lessons)
Decimals and Primes (8 lessons)
-Primes & Composites 
-Primes Factorization
-Greatest Common Divisors & Multiples.
 
-Prime Factorization Aids 
(Learn how to find factors quickly)
-Prime Factorization Examples
 
-Counting & Generating. Factors

-Divisibility Rules and Remainders for Division by 2, 3, 5, 9 and 11.
Integers (12 lessons) Intro to Signed Numbers
Fractions (< 20 lessons)  Essential Skills & Concepts 
Ratios & Fractions (3 lessons):  Similarities & Differences
  
Units in calculations
Fractions  with Units
B.  Basic Algebra
Solving Linear Equations  
- in one unknown. Intro  with stick diagrams?
the normal way
 & with good nttn.
(the nttn that reappears in Gaussian Elimination. |
-in more unknowns: simultaneous equations essentially one unknown. the let algebra do the work view of  word problems.
  - still in more unknowns:  Gaussian Elimination via substitution, by equality or comparison, by operations on equations
C. More Algebra
Words before symbols: See if U like the lengthy chapters 8 to 12 in Volume 2, Three Skills for Algebra  
What is a Variable.  The answer here  is a simple prequel to the modern mathematics viewpoint.
First, every rule & pattern U meet in math, logic & science will be used forwards and backwards.  Get a head start with this theme by reading  Chapter 14 in Three Skills for AlgebraSecond, in the study of Proportionality Relations (3 dense lessons here) finding the proportionality constant gives an initial  backward  use of the proportionality formula.
 Talking about words before symbols and the forward and backward use of formulas gives words to make algebra simpler & clearer.  
If you can not read or write precisely, you will have difficulty in following instructions.  One wordy remedy  is given by chapters 2 to 5  in Three Skills for AlgebraWhere does Logic or a geometric model for reason Appear in Mathematics? The answer lies in  Euclidean-Geometry    In North America, Euclidean Geometry disappeared from high school mathematics as it was too hard. The light treatment here is a possible remedy.
E.  More Geometry
The Pythagorean Theorem. Chapter 17 from  in Three Skills for Algebra uses algebra and geometry   to show why the  Pythagorean equation  for right triangles holds. Its forward and backward use  is common exercise..  At a more theoretical level, the Pythagorean theorem leads the discovery that not all lengths can be  fractional multiples of a unit length. That geometrically implies a  need for and even existence of irrational numbers.
Analytic Geometry:
Common Practices with  Maps and Plans drawn to scale  give coordinate-dependent base  for senior high school development of similarity, trig, vectors and straight lines.   
Complex Numbers: This lesson on
Complex Numbers  draws on Euclidean and Analytic geometry. Sbortcuts simplifiy  trig identities, the cosine law; and   trig formulas for 2D dot- and cross-products. 

F. Logarithms, Exponentials,
Roots & Powers

Logarithms, exponentials, rational and real powers for secondary students. This  complete Operational Viewpoint. (Sufficient for the precalculus forward and backward use of compound growth and decay formulas in biology, physics, chemistry,  personal finance, and calculus. To learn more, if you study calculus,  see chapter 19 of Volume 3, Why Slopes and More.Math

In Volume 2, Three Skills for Algebra, chapters
  1. Geometric Sums Etc,
  2. Notation For Sums,
  3. Personal Money Maths and
  4. Some Finite Mathematics
identify methods useful in money computations, methods needed for calculus. Your teachers or other writer may present the same ideas with greater clarity and detail - A site to do.

H. Polynomial & Quadratics

Analytic Geometry:   -  Slopes and Lines - Take 1.   Take 2 appears in site section Maps and Plans.   Two views are better than one.  I may combine them later.  -In my school days, slopes appeared year after year.   This Why  Slopes calculus preview on graphs of functions y = f(x) explains why.  Enjoy.
Quadratics and Polynomials: Operations on Polynomials:
Meet a light and ultraquick geometric introduction to  multiplication, addition and subtraction of polynomials. Then see how the foregoing combine to permit long division of polynomials.    Compare Fractions  with Units. Enrichment: A Plus:  The Geometric introduction here gives or is almost identical to a justification for column methods in decimal arithmetic. 
Geometric Derivation of the Quadratic Formula  The account here gives a starter lesson for the more algebraically harder geometric-free derivation. If you study physics, chemistry or trigonometry, you will need to know about quadratics, their factorization and the quadratic formula.
Technical Value: The study of polynomials  high school mathematics has technical value as part of the senior high school mathematics preparation for calculus.  This simple account of Why Factor Polynomials   (Chapters 2 to 6 in Volume 3 .Why.Slopes.&.More.Math.) will give a context for the study of polynomials,  their factorization, and sign analysis of functions, all in a way that should improve your algebraic thinking and reasoning skills. 
Vectors in the Plane (2 simple lessons)
- Navigation with vectors or arrows
- Sum of Motions
- more lessons to be added later.
Operations on movement or vectors along the line and in the plane have value in mathematics in defining and implying the properties of real and complex numbers before the assumption of those properties as axioms.  Vectors and their properties appear in physics, its mathematical description and formulation. 
Functions - Forwards & Backwards.  Here is a full technical reference (24 lessons) for use in a calculus or precalculus course as needed. In it, the set viewpoint of functions expression of modern pure mathematics.  comes from the set-based codification and
In the mathematics education reforms of the 1960s in North America, primary and secondary school mathematics were expressed in terms of sets. That expression has now retreated from primary and secondary school texts. But it still lingers on, and can be very useful, a source of clarity and precision, in the situations where it should be retained: Counting with the aid of sets and functions; the description of functions; the high school account of probability theory; and in the discussion or illustration of ideas in logic. 

J. Pre-Calculus Skill Check

Arithmetic Skill Check.  In the calculus courses I taught 1983-89, too many students had weak skills in arithmetic. I would give and carefully correct these exercises to tell students what they needed to review and master.  
-  All the skills and concepts in 
Chapters 1 to 24 or Volume 2, Three Skills for Algebra: Look for those you do not understand and fill the gaps. Do so quickly while balancing this advice with  your other duties.  Good luck.

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